Given ‘X’ to be universe of discourse, A and B are two fuzzy sets with membership function μA(x) and μB(x) then,
Union
The union of two fuzzy sets A and B is a new fuzzy set A ∪ B also on ‘X’ with membership function defined as follow:
Intersection
Intersection of fuzzy sets A & B, is a new fuzzy set A ∩ B also on ‘X’ whose membership function is defined by
Compliment
Compliment of a fuzzy set A is A with membership function
Product of Two Fuzzy Sets
The product of two fuzzy sets A & B is a new fuzzy set A.B with membership function:
Equality
Two fuzzy sets A and B are said to be equal i.e, A = B if and only if μA(x) = μB(x) Which means their membership values must be equal.
Product of Fuzzy Sets with a Crisp Number
Multiplying a fuzzy set A by a crisp number ‘n’ results in a new fuzzy set n.A, whose membership function is
Power of a Fuzzy Set
The alpha power of a fuzzy set A is a new fuzzy set Aα whose membership function is:
that is, individual memberships power of α
Difference of Fuzzy Sets
The differences of two fuzzy sets A and B is a new fuzzy set A-B which is defined as
Disjunctive Sum of A & B
It is the new fuzzy set defined as follow:
A, assuming n is a member of the universal set of discourse, in no way shape or for yields a new set A' with u'(x) = n * u(x). It yields a set A'' with u''(x) = u(x * n). All the multiplication of a crisp function does is rescale (and potentially reverse, given that n is negative) the membership function. You're confusing the set of X with the set of F (fuzzy membership values). Additionally, the product of two fuzzy sets you're describing is actually an intersection operation in which the t-norm utilized is multiplication as opposed to the standard t-norm of min. The actual formula for the product of two fuzzy sets is far more complicated, and (for at least in the case of convex sets, I haven't worked much with non-convex) is u_{A * B}(w) = (for all z in X, sup_{z = x * y}(A(x),B(y)))(w). Your alpha power set is wrong, alpha is traditionally used as a "cutting function" on fuzzy sets, to eliminate support of a fuzzy set that is less than the alpha threshold, i.e. if A has a membership function of u(x), then A' = A^alpha has a membership function u'(x) = {u(x) where u(x) >= alpha, 0 otherwise}. The actual definition of the power of a fuzzy set is analogous to the multiplication operation I described above, but in the cases of even powers the support is usually restricted to positive values of the universe of discourse in order to remain analogous to crisp multiplication. What you're actually describing is a specific class of "linguistic hedges," which are functions on fuzzy sets designed to affect membership in a meaningful way. Your difference of fuzzy sets is correct, with the caveat that you should be calling it the "set difference" and be using the set difference operator as the arithmetic difference has a definition in line with the actual multiplication definition I described earlier, and in fact this arithmetic difference isn't actually defined in situations where the universe of discourse is limited to positive values. This is due to the potential of extending support into an undefined (i.e. negative) region.